Inversion of magnetotelluric data for 2D structure with sharp ...

GEOPHYSICS, VOL. 69, NO. 1 (JANUARY-FEBRUARY 2004); P. 78–86, 11 FIGS., 3 TABLES.10.1190/1.1649377Inversion of magnetotelluric data for 2D structurewith sharp resistivity contrastsCatherine de Groot-Hedlin ∗ and Steven Constable ∗ABSTRACTWe have developed a linearized algorithm to invertnoisy 2-D magnetotelluric data for subsurface conductivitystructures represented by smooth boundaries definingsharp resistivity contrasts. We solve for both a fixednumber of subsurface resistivities and for the boundarylocations between adjacent units. The boundary depthsare forced to be discrete values defined by the mesh usedin the forward modeling code. The algorithm employs aLagrange multiplier approach in a manner similar to thewidely used Occam method. The main difference is thatwe penalize variations in the boundary depths, ratherthan in resistivity contrasts between a large number ofadjacent blocks. To reduce instabilities resulting from thebreakdown of the linear approximation, we allow an optionto penalize contrasts in the resistivities of adjacentunits.We compare this boundary inversion method to thesmooth Occam inversion for two synthetic models, onethat includes a conductive wedge between two resistorsand another that includes a resistive wedge between twoconductors. The two methods give good agreement forthe conductive wedge, but the solutions differ for themore poorly resolved resistive wedge, with the boundaryinversion method giving a more geologically realisticresult. Application of the boundary inversion method tothe resistive Gemini subsalt petroleum prospect in theGulf of Mexico indicates that the shape of this salt featureis accurately imaged by this method, and that themethod remains stable when applied to real data.INTRODUCTIONSince any real magnetotelluric (MT) data set consists of afinite number of imprecise data values, the nonlinear geophysicalinverse problem of finding a model with a response that fitsthe data is ill-posed, and there are an infinite number of modelsthat fit the data equally well above some threshold misfit (belowwhich no models can be found). The task of model constructionthus entails introducing a selection criterion to narrow thesearch down to a class of model having some preferred characteristics.Two-dimensional (2D) MT inverse methods may becategorized in terms of the selection criteria used to choosefrom this infinite set of models.Parameterized inversion methods (e.g., Jupp and Vozoff,1977; Oristaglio andWorthington, 1980) follow the strategy offinding a model that is close to some “ideal” form. Interpretationsbased on this method involve constructing a cross-sectionof the MT transect based on prior geological knowledge,and computing conductivities through least squares inversion.These methods are highly dependent on model parameterizationand on the accuracy and availability of the prior geologicalinformation.Another option is to overparameterize the problem, that is,break the model into more parameters than there are data,and constrain the inversion through a regularization method(Tihonov, 1963a, b). Regularization involves simultaneouslyminimizing the data misfit as well as some other undesirablefeature of the model. A now standard approach is to findthe smoothest model that fits the data. Various methods ofdefining model smoothness may be found in Rodi (1989), deGroot-Hedlin and Constable (1990), Smith and Booker (1991),Uchida (1993), and Rodi and Mackie (2001). Imposition of thesmoothness constraint stabilizes the inversion algorithm andavoids the introduction of anomalous conductivities that aresimply artifacts of the inversion method. However, models derivedby these methods may be geologically unrealistic in situationsin which it is known (from well-log information, forinstance) that the subsurface structure consists of geologicalformations of nearly uniform conductivity separated by sharpboundaries. This is often the case in petroleum exploration targetsin which the MT method is applied (e.g., Hoversten et al.,2000).Given that the MT method involves the measurement of relativelylow-frequency, diffusive electromagnetic (EM) fields,Manuscript received by the Editor November 27, 2002; revised manuscript received July 29, 2003.∗University of California at San Diego, Scripps Institute of Oceanography, 8775 Biological Grade, La Jolla, California 92093-0225. E-mail:chedlin@ucsd.edu; sconstable@ucsd.edu.c○ 2004 Society of Exploration Geophysicists. All rights reserved.78

80 de Groot-Hedlin and ConstableThe n r × (n r + 1) matrix S rho operates only on the resistivityparameters, and is defined by⎛⎞c −c 0 0 ...S rho =0 c −c 0 ...⎜⎝ ...⎟⎠ , (3)... c −cwhere c is a constant that controls the relative weighting betweenlayer roughness and resistivity contrasts. The penaltymatrix S dep acts only on the depth parameters, and is given by⎛⎞S 1 0 ...S dep =0 S 2 0 ...⎜⎝ ...⎟⎠ , (4)... 0 S n rwhere 0 is an (n c − 1) × n c matrix of zeros and S i is the(n c − 1) × n c roughening matrix for the bottom of the ith layer,i.e.,⎛⎞1 −1 0 ...S i =0 1 −1 0 ...⎜⎝ ...⎟⎠ . (5)... 0 1 −1Thus S i differences the model depth parameters between laterallyadjacent columns in layer i.We simultaneously minimize model roughness and data misfitby finding a stationary point for the functionalU =‖Sm‖ 2 + µ −1[ ‖W(d − F[m])‖ 2 − χ 2 ∗]. (6)The trade-off between the data misfit, given by the second term,and the model roughness is controlled by the Lagrange multiplierµ −1 . In equation (6), d represents the magnetotelluricdata, F[m] is a nonlinear forward functional F acting uponthe model m to produce the model response, χ∗ 2 is the desiredlevel of fit, and W is a diagonal matrix with entries inverselyproportional to the data errors.Since the forward functional F[m] is nonlinear with respectto perturbations in resistivity or boundary depth, we linearizeabout a starting model m i . The first two terms of the Taylorapproximation yieldF[m i + m] = F[m i ] + J i m, (7)where J i is the Jacobian matrix of partial derivatives of F[m i ]with respect to the model parameters. The stability of any linearizedinversion method is controlled largely by the accuracyof this approximation. In MT, equation (7) is a good approximationfor models with limited conductivity contrasts; thus,linearized inversion methods tend to be highly robust for inversionsthat limit conductivity variability. With the large conductivitycontrasts that are characteristic of inversions for boundarylocations, this approximation becomes less accurate, andthe iterations may reach local, rather than global, minima inthe solution space. In the next two sections, we discuss methodsto avoid converging to a local minimum.We use the 2D finite element code of Wannamaker et al.(1987) to derive forward responses, and the adjoint method,described in deLugao and Wannamaker (1996), to computepartial derivatives of F[m i ] with respect to conductivity. Use ofthe adjoint method allows us to obtain the model sensitivitieswith respect to conductivity at minimal computational cost.To take advantage of the computational time saving offeredby use of the adjoint method, we use a chain rule method toconvert derivatives with respect to conductivity to derivativeswith respect to depth. This is described in greater detail in theAppendix.As derived in Constable et al. (1987), the model m i+1 is givenby the stationary points of U, i.e.,m i+1 = [ µ(S T S) + (WJ i ) T (WJ i ) ] −1(WJi ) T W ˆd i , (8)where ˆd i is given by d − F[m i ] + J i m i . Note that we need S T S,not S itself. We use sparse matrix multiplication to computeS T S efficiently. Boundary depths computed from equation (8)are adjusted to the value of the nearest block boundary, asindicated in Figure 1, prior to computation of the forward response.Given that model sensitivities with respect to depth aremuch smaller than the derivatives with respect to unit conductivity,this does not significantly alter the model responses. Ateach iteration, we vary µ to find the model which generates theminimum misfit, and use this model as the starting point in thenext iteration. If the misfit is less than χ ∗ , we increase µ to findsmoother models until the desired misfit is attained.Unlike for the Occam method (de Groot-Hedlin and Constable,1990), the starting model for this algorithm must possesssome resistivity contrasts. One reason is that our methodof computing derivatives with respect to resistivity (describedin the Appendix) requires that some conductivity contrast ispresent in adjacent units. Another reason is that we initializeboundary depths in terms of where conductivity contrasts existin the initial model. For subsequent inversions, intermediatemodel solutions that feature very low resistivity contrasts betweenadjacent layers can also yield instability in the algorithm,since the model response is insensitve to perturbations in theboundary depth between two adjacent units with almost identicalresistivity. In this case, the two units can be merged intoone for further iterations. Also note that, because boundariesare parameterized in terms of the logarithm of the depth ratherthan thickness, the solution of equation (8) allows layer boundariesto cross over. When this occurs, we set the depth of the“lower” boundary equal to that of the “upper” boundary, thuspinching out a portion of the intermediate layer. Layers thatare completely pinched out in an inversion must be removedfor further iterations to maintain stability.In application, we found that the algorithm converged slowlyunless we chose a fairly good starting model. One option, usedin the real data example, is to find the best layered model fittingthe data with a 1D inversion algorithm, and use this layeredsolution as a starting model for the 2D algorithm. Anotheroption is to use results from the smooth Occam method tobuild a starting model. We note that different starting modelsmay yield different model solutions, especially when solvingfor a model with many layers. We attribute this to the inherentnonlinearity of the MT problem in the presence of sharpresistivity contrasts; the solution space of a nonlinear problemmay possess many local minima in which a linearized algorithm

MT Inversion for Resistivity Boundaries 81may become trapped. The inversion algorithm is terminatedwhen the inversion models show little variation after severaliterations.SYNTHETIC TESTSWe test the algorithm on two models of the type shown inFigure 2, with resistivity values of each model listed in Table 1.Model A features a conductive wedge underlying a highly resistivesurface layer and overlying a basement of intermediateresistivity, representative of sediments below basalts. The secondmodel (B) represents a resistive wedge-shaped body withina more conductive matrix, typical of salt structures found inthe Gulf of Mexico. For each model, apparent resistivity andphase data for both the transverse electric (TE) and transversemagnetic (TM) modes were generated at 21 stations, spaced at0.5-km intervals. The frequency ranges were chosen such thatthe penetration depth corresponds to the top unit at the highestfrequency and to the bottom unit at the lowest frequency.Responses were computed at five frequencies per decade (i.e.,at 24 frequencies from 400 to 0.01 Hz for the model featuringthe conductive wedge, and for 20 frequencies from 4 to0.0063 Hz for the resistive wedge model). To simulate noisydata, 5% random Gaussian noise was added to the resistivityand 1.4 ◦ to phase data prior to inversion. Since data errorsare Gaussian distributed, and the diagonals of W are the reciprocalsof standard deviations in the apparent resistivity andphase, the desired level of misfit is rms 1.0.For each inversion, the models were represented by a gridof 68 columns and 36 rows of blocks. Although the 5.7 ◦ slopeat the upper boundary of the wedge and the 16.6 ◦ slope at thelower boundary were modeled precisely using triangles in thefinite-element formulation of the forward modeling algorithm,Table 1. Synthetic model resistivities (in ohm-meters).Unit number Model A Model B1 1000 12 10 1003 100 1the inversion procedure only allows for slopes to be modeledusing a stairstep approximation. Furthermore, the grid used forthe inversion is not the same one used for the forward computations;that is, the grid we used does not allow depth values to beassigned their “true” values used in the synthetic model. Thisprovides a realistic test of the algorithm as the locations of conductiveboundaries are rarely known in advance. We computedresponses for models discretized on this grid that most closelyresemble the starting models of Figure 2 and Table 1 to determinethe effect of gridding on the data misfit. We found thatthe rms misfit for the gridded model was 1.55 for the conductivewedge model and 1.5 for the resistive wedge model; thus,a finer grid discretization would be needed to represent themodel more accurately. However, note that these are not necessarilythe best-fitting models for this level of discretization,as resistivities and boundary depths can be adjusted slightly inorder to yield a model that fits the data slightly better.The number of discrete resistivity units must be specifiedin the starting model. We used a three-layer starting modelfor each inversion of the synthetic data, with resistivities anddepths as listed in Table 2. The boundary depth values for thestarting model were set by dividing the model into three layers,each consisting of 12 rows of blocks; initial resistivities werebased on values shown in the relevant pseudosections of theapparent resistivity data.Model A: Conductive wedgeThe noisy synthetic data for model A are shown in pseudosectionin Figure 3. The data were inverted with no penalty onTable 2. Starting model for each inversion.Boundary depth Model A Model B(km) (ohm-m) (ohm-m)1.102 631 1.32.590 31.6 3.2half-space 63.1 1.6FIG. 2. The 2D resistivity model for the inversions of syntheticdata, shown at 2:1 vertical exaggeration. The model featurestwo sloping boundaries and a blunt termination for the middlelayer. Unit resistivities are given in Table 1. Triangles indicatestation locations.FIG. 3. Pseudosections of the responses of model A, whichfeatures a conductive wedge. (a) TE apparent resistivitieslog 10 (ρ a ). (b) TM apparent resistivities log 10 (ρ a ). (c) TE phaseresponses. (d) TM phase responses.

82 de Groot-Hedlin and Constablethe differences in resistivity (i.e., only lateral variability in layerdepths were penalized). The best-fitting model achieved by thisinversion was attained after 13 iterations and has an rms misfitof 1.26. Comparison of the solution (shown in Figure 4) with theoriginal wedge shape shown superimposed on the model, indicatesthat both the unit conductivities and boundary locationshave been well imaged. In particular, the shape of both thetop and bottom wedge boundaries are well recovered withinthe limits afforded by the stairstep approximation. However,instead of a blunt termination of the wedge to the left, the recoveredconductor pinches out gradually, and several isolatedconductors appear to the left of where the wedge terminationshould occur. These artifacts result from the smoothing constraintsforced on the layer-depth parameters, and indicate thatblunt terminations may be poorly imaged by this method.Comparison of the synthetic data and model responses indicatesthat the fit is poorest for the five stations at the left endof the model. The poor fit at these stations is due to the thepresence of the isolated conductors there, which result fromthe smoothing constraint on the layer boundaries. The adequatefit at the remainder of the stations suggests that slopingboundaries are adequately represented by the stairstepapproximation.For comparison, we inverted these data using the Occam 2DMT algorithm (de Groot-Hedlin and Constable, 1990), whichsolves for the smoothest possible model (Figure 5). Again, werepresented the model by a grid of 68 columns and 36 rows.We specified an rms misft equal to that achieved by the boundarylayer inversion. We note that much lower misfits couldhave been achieved by the smooth inversion method, becausethere are (68 × 36) free parameters using this method, whilethe boundary layer inversion has only (3 + 68 × 2) degrees offreedom. The solution to this inversion was attained after sixiterations. The shape of the wedge is reproduced quite well,but the resistivities are less well recovered. Also, it is not clearwhere one would choose to place the lower boundary of thewedge structure.Model B: Resistive wedgeThe noisy synthetic data for model B are shown in pseudosectionin Figure 6. The model response is much weaker for thismodel than for model A, which is expected as the MT method isless sensitive to resistive anomalies than to conductive anomalies.Again, the data were inverted with no penalty on the resistivities,with the starting model listed in Table 2. The truemodel was poorly recovered in a straightforward applicationof the boundary location inversion to these synthetic data (i.e.,although the final misfit was only 1.19, the solution was unrealisticallyrough). Relaxing the target misfit to 1.25 still resulted inan overly rough model, suggesting that the inversion achieveda local, rather than global, minimum. We attribute this to theFIG. 5. Model recovered from smooth inversion of data shownin Figure 3. The model is shown at 2:1 vertical exaggeration. Thewedge boundaries are shown superimposed on the model. Thestation locations are indicated by the triangles at the top ofthe profile.FIG. 4. Model recovered from boundary location inversion ofdata shown in Figure 3. The model is shown at 2:1 vertical exaggeration.The wedge boundaries are shown superimposed onthe model. The station locations are indicated by the trianglesat the top of the profile.FIG. 6. Pseudosections of the responses of model B, which featuresa resistive wedge. (a) TE apparent resistivities log 10 (ρ a ).(b) TM apparent resistivities log 10 (ρ a ). (c) TE phase responses.(d) TM phase responses.

MT Inversion for Resistivity Boundaries 83fact that the inverse problem becomes increasingly nonlinearwith increasing roughness; thus, the Jacobian sensitivities arean inaccurate estimate of the gradient in the solution space.To obtain a better model, we split the inversion into severalsteps in which we sequentially lower the target χ 2 . At each step,we reach the target misfit and perform several more iterationsto smooth the model further. This ensures that the startingmodel at each iteration is reasonably smooth, so that a globallysmooth model can be found. The best smoothed model wecould achieve has an rms misfit of of 1.15 and was reachedafter a total of 22 iterations.This model, shown in Figure 7, shows that both the resistivitiesand boundary locations are reasonably well recovered;the resistivity of the wedge is 74 ohm-m in this model, andthe overlying and underlying layers each have a resistivity of1 ohm-m. The blunt termination of the wedge at the left is accuratelyimaged, but the flattening out of the wedge at the rightis poorly imaged. Indeed, the data residuals indicate that thefit is poorest for the stations at the right.We applied the Occam inversion method to these data, witha target rms of 1.15; the model shown in Figure 8 was attainedafter nine iterations. Obviously, this model does not resemblethe model recovered from the boundary inversion method, althoughthe response yields an equal misfit to the data. This issymptomatic of the nonuniqueness inherent in the MT method.The fact that highly dissimilar models can yield equivalent responsesindicates that the model solution is highly dependenton the choice of norm imposed on the model within the regularizationprocedure. It also indicates the importance of a reasonablechoice of starting model for the sharp boundary inversion.Thus, application of geologically realistic constraintsis particularly important for imaging resistive targets using theMT method.INVERSION OF REAL DATAA marine MT(MMT) survey was carried out in the Gulf ofMexico in 1997 to test the feasibility of using sea-floor MT datato image the base of salt structures (Constable et al., 1998;Hoversten et al., 2000). Imaging these structures is of interestto the petroleum industry as salt structures are associatedwith hydrocarbon traps. The high acoustic contrast between saltbodies and the surrounding matrix makes it difficult and costlyto image the shape of these bodies seismically. On the otherhand, the high resistivity contrast between salt bodies and thesurrounding water-saturated sediments (salt bodies are highlyresistive) makes it a good target for MMT exploration.The 1997 MMT survey was conducted over the Gemini subsaltpetroleum prospect located at 28 ◦ 46 ′ N, 88 ◦ 36 ′ W. Its shapewas previously determined using 3D seismic prestack depth migration,along with depth constraints provided by well logs. Inthis section, we use the boundary inversion method to examinewhether MMT data can be used to independently constrain theshape of a salt body. Bathymetry along the transect consideredhere is minor; there is less than 200-m relief along the line. Furthermore,the skin depths in the conductive sea-floor sedimentsare small, so field distortions due to bathymetric variations arequickly dissipated. Therefore, bathymetric corrections may beneglected in the following analysis.The Gemini salt structure is a 3D body; thus, the electric andmagnetic fields do not decouple into two modes, as they wouldfor an ideal prism. The survey line was along a perpendicularto a subsection of this structure that is 3–4 times longer than itis wide, approximately along its middle. TM mode impedanceswere calculated using the electric (E) field perpendicular to thelong axis of the body (local strike) and magnetic (H) field parallelto the strike. Previous numerical modeling studies (e.g.,Wannamaker et al., 1984) show that the response for a centrallylocated profile across an elongate 3D body agrees withthe TM response for a 2D body with identical cross-section.Thus, we invert the TM mode data to image the cross-sectionof the 3D body. The TM data, shown at left in Figure 9, arerelatively noisy. Error estimates for the apparent resistivitiesand phases are as high as 50%; the minimum error is taken tobe 10%. Data from 15 sites at an average spacing of 0.9 km areconsidered.FIG. 7. Model recovered from boundary location inversion ofdata shown in Figure 6. The model is shown at 2:1 vertical exaggeration.The wedge boundaries are shown superimposed onthe model. The station locations are indicated by the trianglesat the top of the profile.FIG. 8. Model recovered from smooth inversion of data shownin Figure 6. The model is shown at 2:1 vertical exaggeration.The wedge boundaries are shown superimposed on the model.The station locations are indicated by the triangles at the topof the profile.

84 de Groot-Hedlin and ConstableWe started the inversion of these data with a best-fittingseven-layer model determined from a 1D inversion of the data.The resistivities and depths with respect to the sea floor as listedin Table 3. For the first several iterations, no penalty was appliedto resistivity contrasts between layers. A single layer waspinched out in the first iteration and the inversion was restartedwith six units. After several iterations, the models feature largeconductivity contrasts between units, and the algorithm convergespoorly due to breakdowns in the linear approximations.Therefore, the inversion was restarted with a small penalty appliedto differences between the resistivities of adjacent units[i.e., the value c in equation (3) is given a small nonzero value].The final model, attained after a total of 18 iterations has anrms misfit of 1.2 and is shown in Figure 10. The shape of the bodydetermined by 3D seismic prestack migration is shown superimposed.The model results indicate that the shape of the lowersalt boundary is quite well resolved. Further tests with otherstarting models confirmed that the shape of the salt structureis consistently recovered in the inversion for boundary depths;however, the resistivity of this structure is less well resolved.The salt resistivities recovered ranged from 10 to 100 ohm-m.The TM mode response of the model in Figure 10 is shown atright in Figure 9. Comparison with the original data indicatesthat the broad features of this noisy data set are reproduced;the resistive salt body is manifested as a “pull-up” of the lowresistivities in the middle of the TM resistivity data, and as anisolated region of low values in the phase data. Thin resistivelayers at 1 km below the sea floor to the northeast of the saltbody are not represented in the seismic model, but have beenconfirmed by controlled source EM sounding.For comparison, we inverted these data using the Occam inversion,with a target misfit of 1.2. The algorithm converged tothe model shown in Figure 11 in three iterations. The top of thesalt structure is reasonably well resolved by this smooth inversionmethod, but the bottom is very poorly resolved althoughthe residuals indicate that the broad features of the data areadequately recovered. These results indicate that a contrastin resistivities at the top of the salt boundary is required bythe data, but that little contrast is required at the bottomboundary at this misfit level. However, the existence of a sharpTable 3.Boundary depth(km)Starting model for inverting Gemini data.Resistivity(ohm-m)0.446 0.731.023 1.231.918 1.253.298 1.475.424 1.168.704 3.89half-space 5.96FIG. 10. Model recovered from boundary location inversion ofthe Gemini data. The shape of the salt body recovered from3D seismic prestack migration is shown superimposed on themodel. Station locations are indicated by the triangles at thetop of the profile. The station locations are indicated by the trianglesat the top of the profile.FIG. 9. Gemini data. (a) TM mode log 10(ρ a ) data. (b) Pseudosectionof the response of the model shown in Figure 10log 10 (ρ a ). (c) TM mode phase data. (d) Pseudosection of theTM phase response of the model shown in Figure 10.FIG. 11. Model recovered from Occam inversion of Geminidata. The station locations are indicated by the triangles at thetop of the profile.

MT Inversion for Resistivity Boundaries 85conductivity contrast at the bottom boundary is a reasonablegeological constraint. Thus, given the geological constraint thatthe subsurface consists of regions with sharp resistivity contrasts,the inversion for boundary depths yields the most realisticmodel that fits the data at a given misft level. (Note thatthe bottom boundary can be recovered by the smooth Occamroutine by decreasing the misfit, but in this paper we want tocompare models with similar misfits.)DISCUSSION AND CONCLUSIONSGiven that there are a multiplicity of resistivity structuresthat can fit a noisy MT data set equally well, the choice ofMT inversion method should be guided by the type of subsurfacestructure present. For instance, some smooth inversionmethods, such as the Occam method used in the examples, donot discriminate between vertical and horizontal structure, andare thus useful in regions where little is known a priori aboutthe subsurface. The boundary inversion method described herefavors layered structures with possibly large conductivity contrastsbetween adjacent layers, and is useful where the geologicalformations are relatively flat lying.For structures that are very well resolved by the data, likethe conductive wedge, there is little difference in the featuresof the final models obtained using the Occam method and theboundary inversion method. However, a sharp boundary inversionallows depths of interfaces to be found directly withoutinterpreting gradients in resistivity. For structures that arepoorly resolved by the data, like the resistive wedge, the typeof regularization has a much greater impact on the final model.Thus, the boundary inversion method is a good choice if thetarget anomaly is known to be resistive and to have a strongconductivity contrast with the surrounding medium.Due to inaccuracy of the linear approximation in the presenceof strong conductivity contrasts, the boundary inversionmethod can be unstable, and somewhat prone to finding localminima. One solution, used in the application of this methodto the Gemini data set, is to apply a small penalty to the differencesin resistivity between adjacent layers. This prevents thecontrast from becoming large in the initial iterations, so that thelinear approximation remains sufficiently accurate. Anotherapproach, demonstrated in the resistive wedge example, is toapply the boundary inversion method in several steps so thatwe find the smoothest model at a particular misfit at each step.In this way, the algorithm’s search is directed toward modelswith smooth boundaries.ACKNOWLEDGMENTSWe thank Kerry Key for processing the Gemini data anddiscussions on its analysis, David Bartel of Chevron Texaco forproviding us the shape of the Gemini salt volume as determinedby seismic data, and Phil Wannamaker for providing us with theadjoint method of solving for Jacobian sensistivities, which spedup the inversion algorithm considerably. We are also gratefulto Pamela Lezaeta, Colin Farquharson, and an anonymous reviewerfor careful reviews. This work was supported by theScripps Seafloor Electromagnetic Methods Consortium.REFERENCESConstable, S. C., Parker, R. L., and Constable, C. G., 1987, Occam’sinversion: A practical algorithm for generating smooth models fromEM sounding data: Geophysics, 52, 289–300.Constable, S. C., Orange, A., Hoversten, G. M., and Morrison, H. F.,1998, Marine magnetotellurics for petroleum exploration, Part 1: Asea-floor equipment system: Geophysics, 63, 816–825.de Groot-Hedlin, C., and Constable, S., 1990, Occam’s inversion to generatesmooth, two-dimensional models from magnetotelluric data:Geophysics, 55, 1613–1624.de Lugao, P. P., and Wannamaker, P. E., 1996, Calculating the twodimensionalmagnetotelluric Jacobian in finite elements using reciprocity:Geophysical Journal International, 127, 806–810.Gill, P. E., Murray, W., and Wright, M. H., 1981, Practical optimization:Academic Press.Hoversten, G. M., Constable, S. C., and Morrison, H. F., 2000, Marinemagnetotellurics for base-of-salt mapping: Gulf of Mexico field testat the Gemini structure: Geophysics, 65, 1476–1488.Jupp, D. L. B., and Vozoff, K., 1977, Two-dimensional magnetotelluricinversion: Geophysical Journal of the Royal Astronomical Society,50, 333–352.Marcuello-Pascual, A., Kaikkonen, P., and Pous, J., 1992, 2-D Inversionof MT data with a variable model geometry: Geophysical Journal International,110, 297–304.Mehanee, S., and Zhdanov, M., 2002, Two-dimensional magnetotelluricinversion of blocky geoelectrical structures: Journal of GeophysicalResearch, 107, EPM1-11.Oristaglio, M. L., and Worthington, M. H., 1980, Inversion of surfaceand borehole electromagnetic data for two-dimensional electricalconductivity models: Geophysical Prospecting, 28, 633–657.Rodi, W., 1989, Regularization and Backus-Gilbert estimation in nonlinearinverse problems: Applications to magnetotellurics and surfacewaves: Ph.D. dissertation, Pennsylvania State University.Rodi, W., and Mackie, R. L., 2001, Nonlinear conjugate gradients algorithmfor 2-D magnetotelluric inversion: Geophysics, 66, 174–187.Smith, T., Hoversten, M., Gasperikova, E., and Morrison, F., 1999,Sharp boundary inversion of 2D magnetotelluric data: GeophysicalProspecting, 47, 469–486.Smith, T., and Booker, J. R., 1991, Rapid inversion of two- and threedimensionalmagnetotelluric data: Journal of Geophysical Research,96, 3905–3922.Tihonov, A. N., 1963a, Regularization of incorrectly posed problems:Soviet Math. Dokl., 4, 1035–1038.——— 1963b, Solution of incorrectly formulated problems and the regularizationmethod: Soviet Math. Dokl., 4, 1624–1627.Uchida, T., 1993, Smooth 2-D inversion for magnetotelluric data basedon statistical criterion ABIC: Journal of Geomagnetism and Geoelectricity,45, 841–858.Wannamaker, P. E., Hohmann, G. W., and Ward, S. H., 1984, Magnetotelluricresponses of three-dimensional bodies in layered earths:Geophysics, 49, 1517–1533.Wannamaker, P. E., Stodt, J. A., and Rijo, L., 1987, A stable finiteelementsolution for two-dimensional magnetotelluric modeling:Geophysical Journal of the Royal Astronomical Society, 88, 277–296.APPENDIX ACOMPUTATION OF MODEL SENSITIVITIESThe finite element code for 2D MT modeling, described inWannamaker et al. (1987), is used for computation of modelresponses, along with code to calculate Jacobian sensitivities,described by de Lugao and Wannamaker (1996). The modelsensitivities are derived using an adjoint method, which allowsthe sensistivities to be obtained at minimal additionalcomputational cost over that required for calculation of modelresponses.The Jacobian terms output by this code provide partialderivatives of each datum with respect to small changes inmodel resistivities. It is simple to convert these quantities toJacobian sensitivities with respect to log(ρ i ) by application of

86 de Groot-Hedlin and Constablethe chain rule, i.e.,∂(datum)∂(log(ρ i )) = ∂(datum) × ∂(ρ i)∂(ρ i ) ∂(log(ρ i ))= ∂(datum) × 2.3026ρ i , (A-1)∂(ρ i )where ρ i denotes the resistivity of the ith block. The partialderivatives for changes in data with respect to variations in thelog(resistivity) of each large unit are derived by summing theJacobian values of all small blocks within that large unit. Thesequantities are exact to within the accuracy of the finite elementmodeling code.We also use the chain rule to derive the partial derivativesof the data with respect to small changes in boundary depththrough application of the chain rule, i.e.,∂(datum)∂(ln(d i )) = ∂(datum) × ∂(ρ i)∂(ρ i ) ∂(ln(d i )) , (A-2)where d i is the depth parameter of interest. Again, the firstterm on the right side is the model sensitivity derived fromthe adjoint code. The second term is computed using a finitedifferencemethod, i.e., for a given boundary layer depth, we dividethe differences in resistivity across the boundary by the differencein ln(depth). Given that derivatives are accurate onlyfor small incremental variations, we are introducing some errorinto the data sensitivities with respect to depth, especially ifeither the resistivity contrast or the depth increments are large.It is well known that the accuracy of a central finitedifferencescheme is accurate to second order, whereas the accuracyof forward-difference or backward-difference approximationsare accurate only to first order (e.g., Gill et al., 1981).Therefore, we compute data sensitivities with respect to depthusing∂(datum)∂(ln(d i )) = 1 (2J +∂(ρ)ln(d + ) − ln(d i )+ J − ∂(ρ)ln(d − ) − ln(d i )), (A-3)where d i is the depth of boundary for a particular column withina layer and d + and d − represent downward and upward incrementsin depth (depth increases downward), respectively. Thatis, d + is the depth of the bottom of the subunit just below thegiven boundary parameter, and d − is the depth to the top ofthe subunit just above the given boundary parameter. The resistivitycontrast between the units is given by ∂(ρ), and J + andJ − are the model sensitivities derived from the adjoint codefor the model subunits below and above the given boundaryparameter, respectively.